In this problem, we show a property of odd positive integers. We are required to prove that the square of any odd positive integer can always be written in the form 8q + 1 for some integer q. Question Show that the square of an odd positive integer is of the form 8q + 1 […]
Show that the square of an odd positive integer is of the form 8q + 1 for some integer q Read More »
In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can be written only in the forms 5q, 5q + 1, or 5q + 4 for some integer q. Question Prove that the square of any positive integer is
In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can always be written in the form 4q or 4q + 1 for some integer q. Question Prove that the square of any positive integer is of the form
In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can be written in the form 3m or 3m + 1, but never in the form 3m + 2. Question Prove that the square of any positive integer is
In this problem, we prove a property of integers expressed in algebraic form. We are required to show that the square of any positive integer of the form 5q + 1 is again of the same form. Question Prove that the square of any positive integer of the form 5q + 1 is of the
Prove that the square of any positive integer of the form 5q + 1 is of the same form Read More »
In this problem, we study the relationship between two algebraic forms of a positive integer. We are required to prove that every integer of the form 6q + 5 can also be written in the form 3q + 2 for some integer q, but the converse statement is not true. Question Prove that if a
In this problem, we prove a basic divisibility property of integers. We are required to show that for every positive integer n, the expression n³ − n is always divisible by 6. Question For any positive integer n, prove that n³ − n is divisible by 6. Solution Consider the expressionn³ − n. Taking n
For any positive integer n, prove that n³ − n is divisible by 6 Read More »
Prove That the Product of Three Consecutive Positive Integers Is Divisible by 6 Video Explanation Question Prove that the product of three consecutive positive integers is divisible by 6. Solution Step 1: Let the Three Integers Be Let the first positive integer be \(n\), where \(n\) is a positive integer. Then the next two consecutive
Prove that the product of three consecutive positive integers is divisible by 6 Read More »
Prove That the Product of Two Consecutive Positive Integers Is Divisible by 2 Video Explanation Question Prove that the product of two consecutive positive integers is divisible by 2. Solution Step 1: Let the Two Consecutive Positive Integers Be Let the first positive integer be \(n\), where \(n\) is a positive integer. Then the next
Prove that the product of two consecutive positive integers is divisible by 2 Read More »
Prove That One of (a+b)/2 and (a−b)/2 Is Odd and the Other Is Even Video Explanation Question If a and b are two odd positive integers such that a > b, prove that one of the two numbers \(\frac{a+b}{2}\) and \(\frac{a-b}{2}\) is odd and the other is even. Solution Step 1: Express Odd Integers in